Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 2 / c

Assume first that $X_n\stackrel{d}{\to }X$. Because all variables take values in the compact interval $[-C,C]$, each monomial $x^k$ agrees there with a bounded continuous function on $\mathbb{R}$. The bounded moment criterion for weak convergence in this case begins withfor every natural number $k$.

Conversely, suppose all moments converge. Let $f$ be bounded and continuous. By the Weierstrass approximation theorem, for every $\epsilon >0$ there is a polynomial $P$ withMoment convergence implies $\mathbb{E}[P(X_n)]\to \mathbb{E}[P(X)]$, whileTaking the limit superior and then $\epsilon \downarrow 0$ proves $\mathbb{E}[f(X_n)]\to \mathbb{E}[f(X)]$, which is convergence in distribution.